L(s) = 1 | + (−0.766 − 1.32i)2-s + (0.766 − 0.642i)3-s + (−0.673 + 1.16i)4-s + (−1.43 − 0.524i)6-s + 0.532·8-s + (0.173 − 0.984i)9-s + (0.233 + 1.32i)12-s + (−0.766 + 1.32i)13-s + (0.266 + 0.460i)16-s + (−1.43 + 0.524i)18-s + (−0.5 + 0.866i)23-s + (0.407 − 0.342i)24-s + (−0.5 − 0.866i)25-s + 2.34·26-s + (−0.500 − 0.866i)27-s + ⋯ |
L(s) = 1 | + (−0.766 − 1.32i)2-s + (0.766 − 0.642i)3-s + (−0.673 + 1.16i)4-s + (−1.43 − 0.524i)6-s + 0.532·8-s + (0.173 − 0.984i)9-s + (0.233 + 1.32i)12-s + (−0.766 + 1.32i)13-s + (0.266 + 0.460i)16-s + (−1.43 + 0.524i)18-s + (−0.5 + 0.866i)23-s + (0.407 − 0.342i)24-s + (−0.5 − 0.866i)25-s + 2.34·26-s + (−0.500 − 0.866i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5817844596\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5817844596\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 1.87T + T^{2} \) |
| 73 | \( 1 + 1.87T + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.01169495722478337966997379485, −11.58952195766698265233548143128, −10.15804413881525896044634225712, −9.464200926413797679301120810734, −8.625958467544191874718240478068, −7.62153136794229150128735649942, −6.37725458535952693750632654202, −4.20910329994878808624538866646, −2.83838046183866168281928179967, −1.71818216672499630298919976284,
2.93369459737726509413576750423, 4.70521179965447883351608023496, 5.83323076343614519974055294283, 7.19170087334879226206411709511, 8.076759748930071063219585044455, 8.693095989870929723010969814932, 9.897206494457162580832771227658, 10.32137925947366348543295774805, 12.00829910333914853742164124314, 13.31449246953380975095121742173